#214703

order	512
nilpotency class	3
isoclinism	isoclinic class of #214688
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2308
commutator	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂×C₄²	C₂⁴
prime radical	#35534	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

960 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 64

A minimal faithful permutation representation — 4 generators on 32 points:

(1,2)(3,9,4,7)(5,11)(6,10)(8,12,16,13)(14,15)(17,18,20,23,21,24,27,30)(19,22,26,29,25,28,31,32);(1,3,6,4)(2,7,10,9)(5,13,14,12)(8,11,16,15)(19,25)(22,28)(26,31)(29,32);(1,3,6,4)(2,8,10,16)(5,13,14,12)(7,15,9,11)(17,19,21,25)(18,22,24,28)(20,26,27,31)(23,29,30,32);(1,4,6,3)(2,9,10,7)(5,13,14,12)(8,11,16,15)(19,25)(22,28)(26,31)(29,32)

Polycyclic presentation:

2^9|a^2=g,b^2=h,c^2=hi,d^2=h,f^2=h,g^2=i,b^a=be,c^a=cf,d^a=de,e^a=eh,c^b=cei,f^b=fh,g^b=gh,d^c=di,f^c=fh,g^c=gh,g^d=gh

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 121899);